The function $w(z)$ satisfies the differential equation

$$\frac{d^{2} w}{d z^{2}}+p(z) \frac{d w}{d z}+q(z) w=0$$

where $p(z)$ and $q(z)$ are complex analytic functions except, possibly, for isolated singularities in $\overline{\mathbb{C}}=\mathbb{C} \cup\{\infty\}$ (the extended complex plane).

(a) Given equation $(\dagger)$, state the conditions for a point $z_{0} \in \mathbb{C}$ to be

(i) an ordinary point,

(ii) a regular singular point,

(iii) an irregular singular point.

(b) Now consider $z_{0}=\infty$ and use a suitable change of variables $z \rightarrow t$, with $y(t)=w(z)$, to rewrite $(\dagger)$ as a differential equation that is satisfied by $y(t)$. Hence, deduce the conditions for $z_{0}=\infty$ to be

(i) an ordinary point,

(ii) a regular singular point,

(iii) an irregular singular point.

[In each case, you should express your answer in terms of the functions $p$ and $q$.]

(c) Use the results above to prove that any equation of the form ( $\dagger$ ) must have at least one singular point in $\overline{\mathbb{C}}$.